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Multivalued evolution equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1145.35076
The authors prove the existence of integral solutions to the nonlocal Cauchy problem $$u'(t)\in -Au(t)+F(t,u(t)), \quad 0\leq t\leq T;\ u(0)=g(u)$$ in a Banach space $X$, where $A: D(A)\subset X \rightarrow X$ is $m$-accretive and such that $A$ generates a compact semigroup, $F : [0,T] \times X \rightarrow 2^{X}$ has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and $g :C([0,T]; \overline{D(A)})\rightarrow \overline{D(A)}$. The case when $A$ depends on time is also considered. The proof of the basic result is constructed as follows: First, one considers the linear counterpart of the above equation, $u'(t)\in -Au(t)+ f(t)$, on the same interval $I= [0,T]$, with initial condition $u(0)= u_0$. For the linear problem, the integral solution is defined by means of the integral inequality $$\Vert u(t)- x\Vert^2\le\Vert u(s)- x\Vert^2+ 2\int^t_s\langle f(r)- y,u(r)- x\rangle_+\,dr$$ for all $x\in D(A)$, $y\in Ax$ and $u(0)= u_0$, continuous. An integral solution for the nonlinear equation is defined as satisfying the linear equation for some integrable $f(t)\in F(t,u(t))$. Then, using the existence for the linear equation, one shows the existence in the nonlinear case. An application is provided for heat equation.

35K90Abstract parabolic equations
34G25Evolution inclusions
47J35Nonlinear evolution equations
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